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title: "A satellite of mass \\(m\\) travels in a stable circular orbit of radius \\(R\\) with period \\(T\\) about a planet. The gravitational interaction is modeled by an attractive central force of magnitude \\(F(r) = \\dfrac{C}{r^n}\\), where \\(C\\) is a positive constant and \\(n\\) is a real constant satisfying \\(1 < n < 3\\). The gravitational potential energy of the satellite-planet system is defined such that \\(U(r) \\to 0\\) as \\(r \\to \\infty\\). Which of the following expressions represents the total mechanical energy \\(E\\) of the satellite-planet system?"
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url: "https://nerd-notes.com/ubq/124046/"
date_modified: "2026-09-28T13:30:15+00:00"
---

# A satellite of mass \(m\) travels in a stable circular orbit of radius \(R\) with period \(T\) about a planet. The gravitational interaction is modeled by an attractive central force of magnitude \(F(r) = \dfrac{C}{r^n}\), where \(C\) is a positive constant and \(n\) is a real constant satisfying \(1 < n < 3\). The gravitational potential energy of the satellite-planet system is defined such that \(U(r) \to 0\) as \(r \to \infty\). Which of the following expressions represents the total mechanical energy \(E\) of the satellite-planet system?

A satellite of mass \(m\) travels in a stable circular orbit of radius \(R\) with period \(T\) about a planet. The gravitational interaction is modeled by an attractive central force of magnitude \(F(r) = \dfrac{C}{r^n}\), where \(C\) is a positive constant and \(n\) is a real constant satisfying \(1 < n < 3\). The gravitational potential energy of the satellite-planet system is defined such that \(U(r) \to 0\) as \(r \to \infty\). Which of the following expressions represents the total mechanical energy \(E\) of the satellite-planet system?

- **A.** \(-\dfrac{2\pi^2 m R^2}{T^2}\)
- **B.** \(\dfrac{(n-3)\pi^2 m R^2}{(n-1)T^2}\)
- **C.** \(\dfrac{2(n+1)\pi^2 m R^2}{(n-1)T^2}\)
- **D.** \(\dfrac{2(n-3)\pi^2 m R^2}{(n-1)T^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124046/*